Kernel-Size Lower Bounds: The Evidence from Complexity Theory Andrew Drucker IAS Worker 2013, Warsaw Andrew Drucker Kernel-Size Lower Bounds
Part 2/3 Andrew Drucker Kernel-Size Lower Bounds
Note These slides are a slightly revised version of a 3-part tutorial given at the 2013 Workshop on Kernelization (“Worker”) at the University of Warsaw. Thanks to the organizers for the opportunity to present! Preparation of this teaching material was supported by the National Science Foundation under agreements Princeton University Prime Award No. CCF-0832797 and Sub-contract No. 00001583. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Andrew Drucker Kernel-Size Lower Bounds
Outline 1 Introduction 2 OR/AND-conjectures and their use 3 Evidence for the conjectures Andrew Drucker Kernel-Size Lower Bounds
Outline 2 OR/AND-conjectures and their use Andrew Drucker Kernel-Size Lower Bounds
To be proved Evidence for the OR, AND conjectures: Theorem Assume NP � coNP / poly . If L is NP -complete, t ( k ) ≤ poly( k ) , 1 [Fortnow-Santhanam’08] No deterministic poly-time reduction R from OR = ( L ) t ( · ) to any problem can have output size | R ( x ) | ≤ O ( t log t ) . 2 [D.’12] No probabilistic poly-time reduction R from OR = ( L ) t ( · ) , AND = ( L ) t ( · ) to any problem, with Pr[success] ≥ . 99] , can achieve | R ( x ) | ≤ t . Andrew Drucker Kernel-Size Lower Bounds
Background Let’s back up and discuss: What does NP � coNP / poly mean? Why believe it? Andrew Drucker Kernel-Size Lower Bounds
Background: circuits We use ordinary model of Boolean circuits: ∧ , ∨ , ¬ gates, bounded fanin. Say that decision problem L has poly-size circuits, and write L ∈ P / poly, if ∃ { C n : { 0 , 1 } n → { 0 , 1 } } n > 0 : size ( C n ) ≤ poly( n ) , C n ( x ) ≡ L ( x ) . Non-uniform complexity class: def’n of C n may depend uncomputably on n ! Example: if L ⊆ 1 ∗ , then L ∈ P / poly. Also, BPP ⊂ P / poly. Andrew Drucker Kernel-Size Lower Bounds
Background: nondeterminism Recall: decision problem L is in NP if: ∃ poly-time algorithm A ( x , y ) on n + poly( n ) input bits : x ∈ L ⇐ ⇒ ∃ y : A ( x , y ) = 1 . Andrew Drucker Kernel-Size Lower Bounds
Background: nondeterminism Say that decision problem L is in NP / poly if: ∃ poly-sized ckts { C n ( x , y ) } n on n + poly( n ) input bits : x ∈ L n ⇐ ⇒ ∃ y : C n ( x , y ) = 1 . “Non-uniform NP” Andrew Drucker Kernel-Size Lower Bounds
Background: nondeterminism Recall that coNP = { L : L ∈ NP } . Complete problem: UNSAT = { ψ : ψ is unsatisfiable } . coNP / poly = { L : L ∈ NP / poly } . Andrew Drucker Kernel-Size Lower Bounds
Uniform and nonuniform complexity Connect questions about non-uniform computation to uniform questions? Yes! Need a broader view of nondeterminism... Andrew Drucker Kernel-Size Lower Bounds
Games and computation Given a circuit C ( y 1 , y 2 , . . . , y k ) with k input blocks, consider 2-player game where Player 1 wants C → 1, P0 wants C → 0. Take turns setting y 1 , . . . , y k . Andrew Drucker Kernel-Size Lower Bounds
Games and computation Andrew Drucker Kernel-Size Lower Bounds
Games and computation Andrew Drucker Kernel-Size Lower Bounds
Games and computation Andrew Drucker Kernel-Size Lower Bounds
Games and computation Andrew Drucker Kernel-Size Lower Bounds
Games and computation Andrew Drucker Kernel-Size Lower Bounds
Games and computation Define d -ROUND GAME ( ∃ ) as: Input: a d -block circuit C ( y 1 , . . . , y d ). Decide: on 2-player game where P1 goes first, can P1 force a win? ( C = 1) Define complexity class Σ p d as set of languages poly-time (Karp)-reducible to d -ROUND GAME ( ∃ ). “ d th level of Polynomial Hierarchy” Andrew Drucker Kernel-Size Lower Bounds
Games and computation NP = Σ p Σ p d ⊆ Σ p Facts: (“solitaire”); d +1 . 1 Σ p d � = Σ p Common conjecture: for all d > 0 , d +1 . Otherwise we could efficiently reduce a ( d + 1)-round game to an equivalent d -round one, and how the heck do you do that?? Andrew Drucker Kernel-Size Lower Bounds
Games and computation Games allow us to connect uniform and non-uniform complexity questions: Theorem (Karp-Lipton ’82) Suppose NP is in P / poly . Then, for all d > 2 , Σ p d = Σ p 2 . Andrew Drucker Kernel-Size Lower Bounds
Games and computation Games allow us to connect uniform and non-uniform complexity questions: Theorem (Yap ’83) Suppose NP is in coNP / poly . Then, for all d > 3 , Σ p d = Σ p 3 . Andrew Drucker Kernel-Size Lower Bounds
Games and computation So: the assumption NP � coNP / poly can be based on an (easy-to-state, likely) assumption: “One cannot efficiently reduce a 100-round game to an equivalent 3-round game!” Andrew Drucker Kernel-Size Lower Bounds
The minimax theorem An extremely useful tool. Many applications in complexity theory, beginning with [Yao’77]. Gives alternate (but similar) proof of [Fortnow-Santhanam’08] result; seems crucial for best results in [D.’12]. Andrew Drucker Kernel-Size Lower Bounds
The minimax theorem Setting: a 2-player, simultaneous-move, zero-sum game. Players 1, 2 have finite sets X , Y . (“possible moves”) “Payoff function” Val( x , y ) : X × Y → [0 , 1]. Val( x , y ) defines “payoff from P1 to P2,” given moves ( x , y ). (P1 trying to minimize Val( x , y ), P2 trying to maximize) Andrew Drucker Kernel-Size Lower Bounds
The minimax theorem Mixed strategy for P1: A distribution D X over X . (Mixed strategies can be useful...) Minimax thm says: for P1 to do well against all P2 strategies... it’s enough if P1 can do well against any fixed mixed strategy. Andrew Drucker Kernel-Size Lower Bounds
The minimax theorem Theorem (Minimax—Von Neumann) Suppose that for every mixed strategy D Y for P2, there is a P1 move x ∈ X such that E y ∼D Y [Val( x , y )] ≤ α . Then, there is a P1 mixed strategy D ∗ X such that, for all P2 moves y, E x ∼D ∗ X [Val( x , y )] ≤ α . Follows from LP duality theorem. Andrew Drucker Kernel-Size Lower Bounds
Back to business Time to apply these tools. Let’s restate the [Fortnow-Santhanam’08] result. Will switch from k ’s to n ’s... Andrew Drucker Kernel-Size Lower Bounds
Back to business Theorem (FS’08, restated) Let L be an NP -complete language, L ′ another language, and t ( n ) ≤ poly( n ) . Suppose there is a poly-time reduction R ( x ) = R ( x 1 , . . . , x t ( n ) ) taking t ( n ) inputs of length n, and producing output such that [ x j ∈ L ] . R ( x ) ∈ L ′ � ⇐ ⇒ j Suppose too we have the output-size bound | R ( x ) | ≤ O ( t ( n ) log t ( n )) . Then, NP ⊆ coNP / poly . Andrew Drucker Kernel-Size Lower Bounds
Simplifying To ease discussion: L ′ = L ; Assume t ( n ) = n 10 ; Fix Assume � � x 1 , . . . , x n 10 �� � ≡ n 3 . � R � � (No more ideas needed for general case!) Andrew Drucker Kernel-Size Lower Bounds
Proof strategy Recall L is NP-complete. To prove theorem, enough to show that L ∈ coNP / poly , i.e., L ∈ NP / poly . Thus, want to use R to build a non-uniform proof system witnessing membership in L . Andrew Drucker Kernel-Size Lower Bounds
Shadows For x ∈ { 0 , 1 } n , say that x = ( x 1 , . . . , x n 10 ) contains x if x occurs as one of the x j ’s. Define the shadow of x ∈ { 0 , 1 } n by ⊆ { 0 , 1 } n 3 . shadow( x ) := { z = R ( x ) : x contains x } Andrew Drucker Kernel-Size Lower Bounds
Shadows Andrew Drucker Kernel-Size Lower Bounds
Shadows Andrew Drucker Kernel-Size Lower Bounds
Shadows Andrew Drucker Kernel-Size Lower Bounds
Shadows Andrew Drucker Kernel-Size Lower Bounds
Shadows Andrew Drucker Kernel-Size Lower Bounds
Shadows ∈ L is in the shadow of x , then x / ∈ L . Fact: if some z / (by OR-property of R ...) This is our basic form of evidence for membership in L ! ( z will be non-uniform advice...) Andrew Drucker Kernel-Size Lower Bounds
The main claim Claim (FS ’08) There exists a set Z ⊆ L n 3 , with | Z | ≤ poly( n ) , such that for every x ∈ L n , shadow( x ) ∩ Z � = ∅ . Intuition: the massive compression by R = ⇒ some z is the image of many sequences x , hence is in many shadows. Can collect these “popular” z ’s to hit all shadows (of L n ). Claim easily implies L ∈ NP / poly... take Z as non-uniform advice. Andrew Drucker Kernel-Size Lower Bounds
Shadows To prove x ∈ L ... Andrew Drucker Kernel-Size Lower Bounds
Shadows To prove x ∈ L ... nondeterministically choose x ⊃ x and z ∈ Z , and check: Andrew Drucker Kernel-Size Lower Bounds
Shadows To prove x ∈ L ... nondeterministically choose x ⊃ x and z ∈ Z , and check: Conclusion: x ∈ L . Andrew Drucker Kernel-Size Lower Bounds
The main claim Claim (FS ’08) There exists a set Z ⊆ L n 3 , with | Z | ≤ poly( n ) , such that for every x ∈ L n , shadow( x ) ∩ Z � = ∅ . Andrew Drucker Kernel-Size Lower Bounds
Recommend
More recommend
